[v9.3.x] Docs: Update publish.yml for 9.3.x (#59537)

Update publish.yml for 9.3.x

.github/workflows/publish.yml

@@ -3,7 +3,7 @@ name: publish_docs
 on:
   push:
     branches:
-    - main
+    - v9.3.x
     paths:
     - 'docs/sources/**'
     - 'packages/grafana-*/**'
@@ -17,7 +17,7 @@ jobs:
     - uses: actions/checkout@v3
     - run: git clone --single-branch --no-tags --depth 1 -b master https://grafanabot:${{ secrets.GH_BOT_ACCESS_TOKEN }}@github.com/grafana/website-sync ./.github/actions/website-sync
     - name: setup node
-      uses: actions/setup-node@v3.5.1      
+      uses: actions/setup-node@v3.5.1
       with:
         node-version: '16'
     - name: Get yarn cache directory path
@@ -29,7 +29,7 @@ jobs:
         key: yarn-${{ hashFiles('**/yarn.lock') }}
         restore-keys: |
             yarn-
-    - run: yarn install --immutable    
+    - run: yarn install --immutable
     - name: publish-to-git
       uses: ./.github/actions/website-sync
       id: publish
@@ -39,7 +39,7 @@ jobs:
         host: github.com
         github_pat: '${{ secrets.GH_BOT_ACCESS_TOKEN }}'
         source_folder: docs/sources
-        target_folder: content/docs/grafana/next
+        target_folder: content/docs/grafana/latest
         allow_no_changes: 'true'
     - shell: bash
       run: |